57.1.19 problem 19
Internal
problem
ID
[9003]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
19
Date
solved
:
Sunday, March 30, 2025 at 01:58:33 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} c y^{\prime }&=a x +b y^{2} \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 75
ode:=c*diff(y(x),x) = a*x+b*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\frac {b a}{c^{2}}\right )^{{1}/{3}} \left (\operatorname {AiryAi}\left (1, -\left (\frac {b a}{c^{2}}\right )^{{1}/{3}} x \right ) c_1 +\operatorname {AiryBi}\left (1, -\left (\frac {b a}{c^{2}}\right )^{{1}/{3}} x \right )\right ) c}{b \left (c_1 \operatorname {AiryAi}\left (-\left (\frac {b a}{c^{2}}\right )^{{1}/{3}} x \right )+\operatorname {AiryBi}\left (-\left (\frac {b a}{c^{2}}\right )^{{1}/{3}} x \right )\right )}
\]
✓ Mathematica. Time used: 0.192 (sec). Leaf size: 437
ode=c*D[y[x],x]==a*x+b*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {c \left (x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )} \\
y(x)\to -\frac {c \left (x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a*x - b*y(x)**2 + c*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded